We apply Theorem REF  to the restricted 4-body problem. Denote the eigenvalues of \(D\)  by \({\lambda }_3\)  and \({\lambda }_4\) . By Proposition REF , we have that both \({\lambda }_3\)  and \({\lambda }_4\)  are positive and \({\lambda }_3 + {\lambda }_4 = 3\) .
By this property, we show the relationship between the Maslov index of \(\xi _{{\alpha }, e}\)  given by (REF ) and the Maslov index of \({\gamma }_{\beta , e}\)  which is the elliptic Lagrangian solutions in [1]}.
Since the elliptic Lagrangian solutions has already been well-studied in [1]},
we have the Maslov index of \(\xi _{{\alpha }, e}\)  in Proposition REF .
By the properties of the Maslov index, we define three curves \({\alpha }_k(e)\) , \({\alpha }_s(e)\)  and \({\alpha }_m(e)\)  (cf. (REF ), (REF ) below) from left to right in \(({\alpha }, e)\in [0,3]\times [0, 1)\)  according to the \(\omega \) -Maslov indices of \(\xi _{{\alpha },e}\) .
By the \({\omega }\) -Maslov index theory [3]}, we obtain the normal forms of \(\xi _{a,e}(2\pi )\)  and then the linear stability as follows.
